Mixed Discrete Least Squares Meshfree method for solving the incompressible Navier–Stokes equations

Abstract

A Mixed Discrete Least Squares Meshfree (MDLSM) method is proposed in this paper for the solution of incompressible Navier–Stokes equations. A semi-incremental two-step fractional projection method is first used to discretize the incompressible Navier–Stokes equations, followed by a mixed formulation used to solve the pressure equations. Using the mixed formulation, it is expected that the accuracy of the pressure approximation and in particular the pressure gradients are improved compared with that of conventional solution methods and in particular Discrete Least Squares Meshfree (DLSM) method. DLSM method is based on minimizing the least squares functional defined as the weighted summation of the squared residuals of the differential equation and its boundary conditions. The method is not subject to the Ladyzenskaja–Babuska–Brezzi (LBB) condition since it formulates the problem in the form of a minimization problem rather than a saddle-point problem. A number of numerical experiments are used to evaluate the efficiency of the proposed MDLSM method and to compare its accuracy against the DLSM method. From the results, it is found that the proposed MDLSM method can efficiently simulate the incompressible fluid flow problems. Furthermore, it can be concluded that the MDLSM method has higher accuracy compared with the DLSM method.